Nonparametric Statistic and the Runs Test for Stationarity - Signal Processing of Random Physiological Signals - page 63

for the data under investigation to be free of periodicities, but the fundamental period
must be short compared to the averaging time used to compute sample values.
The number of runs that occur in a sequence of observations gives an indication
as to whether or not results are independent random observations of the same random
variables. Specifically, if a sequence of N observations is independent observations of the
same random variable, that is, the probability of a ( ) result does not change from

one observation to the next, then the sampling distribution of the number of runs in the
sequence is a random variable r with a mean value as shown in (8.4) and the variance is
given by (8.5).

+ )or( −

2 N 1 N 2
N

μ = + 1 (8.4)

r

where N 1 is the number of ( + ) observations and N 2 is the number of ( − ) observations.

2 N 1 N 2 (2 N 1 N 2 − N )

2
r

σ = (8.5)

N 2 ( N 1

− 1 )

For the case where N 1 = N 2 = N / 2 then the mean and variance quations become (8.6

and 8.7.)

N
2 +

μ = 1 (8.6)

r

and

N ( N − 2)

2
r

σ = (8.7)

4 ( N − 1 )

There distribution function is given by Table 8.2.

N
2 =

− 2 r n ; 2 r η is, where η = N 1 = N 2 , r η ;1 and

In any case, a sequence of plus and minus observations is obtained from which
the number of “RUNS” is determined. A “RUN” is defined as a sequence of identical
observations that is followed and preceded by a different observation or no observation at
all. There are tables for the distribution function of runs. Runs test can be used to evaluate
data involving the question of testing a single sequence of observations for independence.